[FOM] Re: The Myth of Hypercomputation
aatu.koskensilta at xortec.fi
Tue Feb 17 02:02:23 EST 2004
Piyush P Kurur wrote:
>On Tue, Feb 10, 2004 at 10:11:06PM +0000, Toby Ord wrote:
> For a "hyper computer" built using a theory say T, one first needs to
>be confident that T is indeed true. For this to be the case, T has to be
>experimentally verified. Suppose that T predicts a fundamental constant
>to have the value $\Omega$ (the halting set coded as a real number), how
>are we going to check it. We consider T to be true iff apart from being
>mathematically consistent should agree to all experiments. All
>predictions made by this theory should be in principle verified.
Why? Of course, if T is true, then all predictions are in "principle"
verified. The physical theories we have now are not verified in practice
to such an extent. The ideal of truth provides guidelines for research,
but "probable" truth or high estimated truthlikenss or what you have is
in practice what is needed for a theory to be accepted - tentatively -
It's entirely possible that a scenario on the lines of following should
happen. Assume that some physical theory T becomes accepted - say T is
some form of quantum gravity or string theory or whatnot - and someone
finds out that some physical set up corresponds to a mechanism
which decides Pi-1 sentences *and* gives counterexamples when the
sentence is not true. We could well have as much confidence in T as in
any of our now common theories. In addition, we could test the
predictions about Pi_1 sentences we know to be true, and some we know to
be false, and see that the results make sense. We could now go
trough interesting undecided Pi-1 sentences until we find one that is
untrue, and is proven untrue by the counterexample the physical
mechanism gives us. If T also gives us a good idea as to why the
mechanism works, I don't think it would be sensible to worry
about philosophical problems about underdetermination of the truth of T
by the finite number of experiments we have tried out, any more than it
is now for quantum physics, general relativity, etc.
Of course, if this situation should happen, it would only be of interest
if we can actually construct such mechanisms, and if the counterexamples
were of feasible size (so that we can for example check them on a
computer). Uncomputability "round the corner" in some remote part and
exotic part of the universe could very well be just an "ideal element"
of the theory in the Hilbertian sense.
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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